Problem:
Amelia needs to estimate the quantity , where , and are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of ?
Answer Choices:
A. She rounds all three numbers up.
B. She rounds and up, and she rounds down.
C. She rounds and up, and she rounds down.
D. She rounds up, and she rounds and down.
E. She rounds up, and she rounds and down.
Solution:
As long as and and their rounded values are positive, rounding the dividend up in a division problem makes the answer larger, and rounding down makes the answer smaller. Similarly, rounding the divisor up makes the answer smaller, and rounding down makes the answer larger. In a subtraction problem , rounding up or rounding down increases the answer, and rounding down or rounding up decreases it. Only in choice (D) do all the roundings contribute to increasing the answer. In the other situations, the estimate may be larger or smaller than the exact value, depending on the the amount by which each number is rounded and their values. In particular, the rounding may make the answer smaller.
For (A), .
For (B), .
For (C), .
For (E), .
The problems on this page are the property of the MAA's American Mathematics Competitions